High-Fidelity loudspeaker systems are generally realized by dividing the audible frequency spectrum into two or more frequency bands and then applying each of these bands to appropriate individual loudspeaker drivers. Frequency band division is typically accomplished by so-called crossover network filters. By using appropriately designed crossover network filters, each driver then can be optimized to best reproduce a particular predetermined range of frequencies for which it is designed. When combined in a loudspeaker system, drivers and crossover filters together comprise a speaker system. Such speaker systems are then capable of more accurately reproducing the entire audible frequency range.
Crossover filters belong to three general types: (1) low-pass filter (LPF) typically used for “woofers” (low frequency drivers); (2) band-pass filters (BPF) used with midrange drivers middle frequencies (e.g., the approximate range of a human voice); and (3) high-pass filters (HPF) typically used with so called “tweeter” drivers (high frequencies typically to approximately 20 KHz).
A woofer and tweeter together with their respective LPF and HPF comprise a so-called “two-way” loudspeaker system well-known to those of skill in the art. In a two-way system, the woofer is typically adapted to provide an upper frequency response extending into to the lower portion of midrange frequencies and possibly a tweeter having a low frequency response extending down into the upper portion of the midrange frequencies. A block diagram of a two-way speaker system of the prior art is shown in FIG. 1b. 
Adding a midrange driver with its respective BPF to a two-way system creates a so-called “three-way” loudspeaker system also well-known to those of skill in the art. The midrange driver with its associated BPF reproduces a predetermined portion of the midrange frequency spectrum, thereby relieving the woofer and tweeter from needing to do so. A block diagram representation of a three-way speaker system is shown in FIG. 2b. 
However, as speaker systems were realized using the crossover networks disclosed in the inventors '389 patent, a slight emphasis of the sound were sometimes observed at frequencies at or near the crossover frequencies. These emphases were both seen in frequency response measurements and were heard in listening sessions.
The Joseph Audio “Pearl”™ loudspeaker has been widely demonstrated many times and has received high praise. At the Consumer Electronics Show (CES) in Las Vegas, January, 2014, the Pearl™ speaker received the “Best Sound in Show” award. There has been many other high honors granted including, for example, at the January 2017 CES as well as other honors given to them at the Rocky Mountain Audio Fest over a period of several years. An Internet search will immediately provide documentation of the accolades received for Joseph Audio Pears™ as well as other speaker models from Joseph Audio using crossover networks designed by the present inventor.
The present inventor had designed a novel, asymmetrical “infinite slope” crossover for the Pears™. Over the course of the life of the Pearl™ speakers, some reviewers (as well as the inventor) noted unusual slight but audible coloration in the area of the crossover frequency, typically at or near the 2 kHz ⅓ octave band. The present inventor pondered what could possibly be causing this phenomenon. He deduced that a localized phase error might possibly be responsible. His intuition caused him to initially discard that theory.
Several years passed and the inventor designed a new crossover for a new Joseph Audio product, the “Perspective Loudspeaker” introduced in May, 2011. Noted reviewer John Atkinson of Stereophile Magazine reviewed the “Perspective” in the July, 2014 issue of Stereophile. John's review included detailed precision measurements that he reported in his review:                “The low-frequency, ⅓-octave band warble tones on my Editor's Choice (CD, Stereophile STPH016-2) were reproduced with full weight down to the 25 Hz band, but with the 32 Hz warble tone significantly reinforced by a room mode. The 20 Hz warble tone was inaudible at normal listening levels. With the half step-spaced toneburst track on the Editor's Choice, the lower frequency tones spoke relatively clearly, but there was some emphasis of the upper-frequency tones in the octave below 4.2 kHz. This presence-region emphasis could also be heard with the dual-mono pink noise from Editor's Choice though the Perspective's reproduction of this signal was otherwise seamless through quite a large vertical window, as long as I didn't stand up. The central image of the pink noise was appropriately narrow and well defined.” [Emphasis added] [Stereophile Magazine, Volume 37, No. 7, pp 62-71]This measurement by John Atkinson rekindled the inventor's curiosity regarding the cause of this anomaly. Having never forgotten his theory that some unusual phase error was occurring at the crossover frequency, the inventor undertook a series of experiments to prove or disprove that theory.        
It would, therefore, be advantageous to provide improved crossover networks for multi-driver speaker systems that maintained a substantially uniform frequency and phase response at or near all crossover frequencies.